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So, let's suppose we're in a 3D space with two points (corners) as the boundaries.

enter image description here

Now, we already know how we can get the easiest path in the 3D space. The problem I faced is how can we get a curve path (Points) that'll connect the both corners while spreading itself (line/path) acquring the most volume in the space.

Let's say we have a function f(A) in which A is the number of points (input) and returns a array of size A consisting of the points which can acquire the most volume in the space in the mean time that can join the starting point and ending point as of a smooth curve (Spline let's say for now)

Suppose we have starting point [0,0,0] and [2,2,2] as ending point. When we pass 1 as A for f(A) it'll return us [1, 1/2, 0] as it'll create a curve in the mean time joining both points and acquiring the most area. Resulting in similar to following point enter image description here

daysling
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  • [Catmull-Rom splines](https://en.wikipedia.org/wiki/Centripetal_Catmull%E2%80%93Rom_splines) are a simple way to connect three points smoothly, but the "most area" is not well-defined since there is no upper-bound. – BlueRaja - Danny Pflughoeft Apr 17 '22 at 08:02
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    most volume? what do you mean by that ? something like [Hilbert curve](https://stackoverflow.com/a/50488348/2521214)? ... is you just want curve going through your points use piecewise interpolation cubics or Catmull-ROM SPLINEs ... You can even [convert your points into BEZIER control points](https://stackoverflow.com/a/22582447/2521214) if your rendering engine has BEZIER implemented ... – Spektre Apr 17 '22 at 08:55
  • Let's just say we've boundries for a curve, I want to get the most volume that can be created by joining the A amount of points – daysling Apr 17 '22 at 11:33
  • @daysling that is too vague ... you can join any 2 points with spiral covering surface of sphere with infinite radius to cover infinite bounding volume but I doubt that is what you want .... so what exactly is your input and curve properties? All the smooth curves like BEZIER,SPLINE,CATMULL-ROM has nothing to do with bounding volume ... maybe image of desired result would help as I got the feeling you chose misleading words and want something entirely else then described – Spektre Apr 17 '22 at 15:01

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